Identifier
Values
[1] => [1,0] => [1] => 1
[1,1] => [1,0,1,0] => [1,2] => 1
[2] => [1,1,0,0] => [2,1] => 1
[1,1,1] => [1,0,1,0,1,0] => [1,2,3] => 1
[1,2] => [1,0,1,1,0,0] => [1,3,2] => 2
[2,1] => [1,1,0,0,1,0] => [2,1,3] => 1
[3] => [1,1,1,0,0,0] => [3,1,2] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 1
[1,1,2] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 3
[1,2,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 2
[1,3] => [1,0,1,1,1,0,0,0] => [1,4,2,3] => 3
[2,1,1] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 1
[2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 3
[3,1] => [1,1,1,0,0,0,1,0] => [3,1,2,4] => 1
[4] => [1,1,1,1,0,0,0,0] => [4,1,2,3] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => 6
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 8
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 4
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 3
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => 6
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => 4
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => 1
[5] => [1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => 10
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 15
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => 6
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => 10
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 10
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 8
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => 20
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => 15
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,2,3,4,6] => 4
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,2,3,4,5] => 5
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => 5
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => 4
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,4,5] => 10
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 3
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => 15
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,3,4,6] => 6
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,3,4,5] => 10
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [3,1,2,4,5,6] => 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [3,1,2,4,6,5] => 5
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,1,2,5,4,6] => 4
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,1,2,6,4,5] => 10
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,1,2,3,5,6] => 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,1,2,3,6,5] => 5
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,1,2,3,4,6] => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,1,2,3,4,5] => 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => 6
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,4,6,5,7] => 5
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,2,4,5,6,7] => 2
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [1,5,2,3,4,7,6] => 24
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,7,2,3,4,5,6] => 6
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6,7] => 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [2,1,3,4,7,5,6] => 15
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [2,1,3,7,4,5,6] => 20
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [2,1,6,3,4,5,7] => 10
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,1,7,3,4,5,6] => 15
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [3,1,2,4,5,6,7] => 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [3,1,2,4,5,7,6] => 6
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [4,1,2,3,5,6,7] => 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [4,1,2,3,5,7,6] => 6
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [5,1,2,3,4,6,7] => 1
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [5,1,2,3,4,7,6] => 6
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6,1,2,3,4,5,7] => 1
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Description
The number of reduced Kogan faces with the permutation as type.
This is equivalent to finding the number of ways to represent the permutation $\pi \in S_{n+1}$ as a reduced subword of $s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n)$, or the number of reduced pipe dreams for $\pi$.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Map
bounce path
Description
The bounce path determined by an integer composition.